A033275 -id:A033275 - cp's OEIS frontend

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

1, 1, 2, 1, 5, 5, 1, 9, 21, 14, 1, 14, 56, 84, 42, 1, 20, 120, 300, 330, 132, 1, 27, 225, 825, 1485, 1287, 429, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 1, 54, 936, 7644, 34398, 91728, 148512, 143208, 75582, 16796

Offset: 3

N. J. A. Sloane

T(n+3, k) is also the number of compatible k-sets of cluster variables in Fomin and Zelevinsky's cluster algebra of finite type A_n. Take a row of this triangle regarded as a polynomial in x and rewrite as a polynomial in y := x+1. The coefficients of the polynomial in y give a row of the triangle of Narayana numbers A001263. For example, x^2 + 5*x + 5 = y^2 + 3*y + 1. - Paul Boddington, Mar 07 2003
Number of standard Young tableaux of shape (k+1,k+1,1^(n-k-3)), where 1^(n-k-3) denotes a sequence of n-k-3 1's (see the Stanley reference).
Number of k-dimensional 'faces' of the n-dimensional associahedron (see Simion, p. 168). - Mitch Harris, Jan 16 2007
Mirror image of triangle A126216. - Philippe Deléham, Oct 19 2007
For relation to Lagrange inversion or series reversion and the geometry of associahedra or Stasheff polytopes (and other combinatorial objects) see A133437. - Tom Copeland, Sep 29 2008
Row generating polynomials 1/(n+1)*Jacobi_P(n,1,1,2*x+1). Row n of this triangle is the f-vector of the simplicial complex dual to an associahedron of type A_n [Fomin & Reading, p. 60]. See A001263 for the corresponding array of h-vectors for associahedra of type A_n. See A063007 and A080721 for the f-vectors for associahedra of type B and type D respectively. - Peter Bala, Oct 28 2008
f-vectors of secondary polytopes for Grobner bases for optimization and integer programming (see De Loera et al. and Thomas). - Tom Copeland, Oct 11 2011
From Devadoss and O'Rourke's book: The Fulton-MacPherson compactification of the configuration space of n free particles on a line segment with a fixed particle at each end is the n-Dim Stasheff associahedron whose refined f-vector is given in A133437 which reduces to A033282. - Tom Copeland, Nov 29 2011
Diagonals of A132081 are rows of A033282. - Tom Copeland, May 08 2012
The general results on the convolution of the refined partition polynomials of A133437, with u_1 = 1 and u_n = -t otherwise, can be applied here to obtain results of convolutions of these polynomials. - Tom Copeland, Sep 20 2016
The signed triangle t(n, k) =(-1)^k* T(n+2, k-1), n >= 1, k = 1..n, seems to be obtainable from the partition array A111785 (in Abramowitz-Stegun order) by adding the entries corresponding to the partitions of n with the number of parts k. E.g., triangle t, row n=4: -1, (6+3) = 9, -21, 14. - Wolfdieter Lang, Mar 17 2017
The preceding conjecture by Lang is true. It is implicit in Copeland's 2011 comments in A086810 on the relations among a gf and its compositional inverse for that entry and inversion through A133437 (a differently normalized version of A111785), whose integer partitions are the same as those for A134685. (An inversion pair in Copeland's 2008 formulas below can also be used to prove the conjecture.) In addition, it follows from the relation between the inversion formula of A111785/A133437 and the enumeration of distinct faces of associahedra. See the MathOverflow link concernimg Loday and the Aguiar and Ardila reference in A133437 for proofs of the relations between the partition polynomials for inversion and enumeration of the distinct faces of the A_n associahedra, or Stasheff polytopes. - Tom Copeland, Dec 21 2017
The rows seem to give (up to sign) the coefficients in the expansion of the integer-valued polynomial (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/(n!*(n+1)!) in the basis made of the binomial(x+i,i). - F. Chapoton, Oct 07 2022
Chapoton's observation above is correct: the precise expansion is (x+1)*(x+2)^2*(x+3)^2*...*(x+n)^2*(x+n+1)/ (n!*(n+1)!) = Sum_{k = 0..n-1} (-1)^k*T(n+2,n-k-1)*binomial(x+2*n-k,2*n-k), as can be verified using the WZ algorithm. For example, n = 4 gives (x+1)*(x+2)^2*(x+3)^2*(x+4)^2*(x+5)/(4!*5!) = 14*binomial(x+8,8) - 21*binomial(x+7,7) + 9*binomial(x+6,6) - binomial(x+5,5). - Peter Bala, Jun 24 2023

			The triangle T(n, k) begins:
n\k  0  1   2    3     4     5      6      7     8     9
3:   1
4:   1  2
5:   1  5   5
6:   1  9  21   14
7:   1 14  56   84    42
8:   1 20 120  300   330   132
9:   1 27 225  825  1485  1287    429
10:  1 35 385 1925  5005  7007   5005   1430
11:  1 44 616 4004 14014 28028  32032  19448  4862
12:  1 54 936 7644 34398 91728 148512 143208 75582 16796
... reformatted. - _Wolfdieter Lang_, Mar 17 2017

S. Devadoss and J. O'Rourke, Discrete and Computational Geometry, Princeton Univ. Press, 2011 (See p. 241.)
Ronald L. Graham, Donald E. Knuth, Oren Patashnik, Concrete Mathematics, 2nd ed., Addison-Wesley, 1994. Exercise 7.50, pages 379, 573.
T. K. Petersen, Eulerian Numbers, Birkhauser, 2015, Section 5.8.

Vincenzo Librandi, Table of n, a(n) for n = 3..2000
Paul Barry, On Integer-Sequence-Based Constructions of Generalized Pascal Triangles, Journal of Integer Sequences, Vol. 9 (2006), Article 06.2.4.
Paul Barry, Continued fractions and transformations of integer sequences, JIS 12 (2009) 09.7.6.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, On the f-Matrices of Pascal-like Triangles Defined by Riordan Arrays, arXiv:1805.02274 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
Karin Baur and P. P. Martin, The fibres of the Scott map on polygon tilings are the flip equivalence classes, arXiv:1601.05080 [math.CO], 2016.
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, 105 (1998), 256-257.
William Butler, Andrew Kalotay and N. J. A. Sloane, Correspondence, 1974
Arthur Cayley, On the partitions of a polygon, Proc. London Math. Soc., 22 (1891), 237-262 = Collected Mathematical Papers. Vols. 1-13, Cambridge Univ. Press, London, 1889-1897, Vol. 13, pp. 93ff. (See p. 239.)
Adrian Celestino and Yannic Vargas, Schröder trees, antipode formulas and non-commutative probability, arXiv:2311.07824 [math.CO], 2023.
Frédéric Chapoton, Enumerative properties of generalized associahedra, Séminaire Lotharingien de Combinatoire, B51b (2004), 16 pp.
Johann Cigler, Some remarks and conjectures related to lattice paths in strips along the x-axis, arXiv:1501.04750 [math.CO], 2015.
Manosij Ghosh Dastidar and Michael Wallner, Bijections and congruences involving lattice paths and integer compositions, arXiv:2402.17849 [math.CO], 2024. See p. 16.
Jesús A. De Loera, Jörg Rambau, and Francisco Santos Leal, Triangulations of Point Sets [From Tom Copeland Oct 11 2011]
Satyan L. Devadoss, Combinatorial Equivalence of Real Moduli Spaces, Notices Amer. Math. Soc. 51 (2004), no. 6, 620-628.
Satyan L. Devadoss and Ronald C. Read, Cellular structures determined by polygons and trees, arXiv/0008145 [math.CO], 2000. [From Tom Copeland Nov 21 2017]
Anton Dochtermann, Face rings of cycles, associahedra, and standard Young tableaux, arXiv preprint arXiv:1503.06243 [math.CO], 2015.
Brian Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
Cassandra Durell and Stefan Forcey, Level-1 Phylogenetic Networks and their Balanced Minimum Evolution Polytopes, arXiv:1905.09160 [math.CO], 2019.
P. Flajolet and M. Noy, Analytic Combinatorics of Non-crossing Configurations, Discrete Math., 204, 1999, 203-229.
Sergey Fomin and Nathan Reading, Root systems and generalized associahedra, Lecture notes for IAS/Park-City 2004, arXiv:math/0505518 [math.CO], 2005-2008. [From _Peter Bala_, Oct 28 2008]
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, arXiv:math/0104151 [math.RT], 2001.
Sergey Fomin and Andrei Zelevinsky, Cluster algebras I: Foundations, J. Amer. Math. Soc. 15 (2002) no.2, 497-529.
Sergey Fomin and Andrei Zelevinsky, Y-systems and generalized associahedra, Ann. of Math. (2) 158 (2003), no. 3, 977-1018.
Ivan Geffner and Marc Noy, Counting Outerplanar Maps, Electronic Journal of Combinatorics 24(2) (2017), #P2.3.
Rijun Huang, Fei Teng, and Bo Feng, Permutation in the CHY-Formulation, arXiv:1801.08965 [hep-th], 2018.
Germain Kreweras, Sur les partitions non croisées d'un cycle, (French) Discrete Math. 1 (1972), no. 4, 333--350. MR0309747 (46 #8852).
Germain Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973.
Germain Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30.
Germain Kreweras, Les préordres totaux compatibles avec un ordre partiel, Math. Sci. Humaines No. 53 (1976), 5-30. (Annotated scanned copy)
Thibault Manneville and Vincent Pilaud, Compatibility fans for graphical nested complexes, arXiv:1501.07152 [math.CO], 2015.
Sebastian Mizera, Combinatorics and topology of Kawai-Lewellen-Tye relations J. High Energy Phys. 2017, No. 8, Paper No. 97, 54 p. (2017).
J.-C. Novelli and J.-Y. Thibon, Hopf Algebras of m-permutations,(m+1)-ary trees, and m-parking functions, arXiv:1403.5962 [math.CO], 2014.
Vincent Pilaud, Brick polytopes, lattice quotients, and Hopf algebras, arXiv:1505.07665 [math.CO], 2015.
Vincent Pilaud and V. Pons, Permutrees, arXiv:1606.09643 [math.CO], 2016-2017.
Ronald C. Read, On general dissections of a polygon, Aequat. Math. 18 (1978), 370-388.
Dean Rubine, Exercises for A Hyper-Catalan Series Solution to Polynomial Equations, and the Geode, arXiv:2507.13045 [math.CO], 2025. See p. 9.
Rodica Simion, Convex Polytopes and Enumeration, Adv. in Appl. Math. 18 (1997) pp. 149-180.
Richard P. Stanley, Polygon dissections and standard Young tableaux, J. Comb. Theory, Ser. A, 76, 175-177, 1996.
Rekha R. Thomas, Lectures in Geometric Combinatorics [_Tom Copeland_, Oct 11 2011]

Cf. diagonals: A000012, A000096, A033275, A033276, A033277, A033278, A033279; A000108, A002054, A002055, A002056, A007160, A033280, A033281; row sums: A001003 (Schroeder numbers, first term omitted). See A086810 for another version.
A007160 is a diagonal. Cf. A001263.
With leading zero: A086810.
Cf. A019538 'faces' of the permutohedron.
Cf. A063007 (f-vectors type B associahedra), A080721 (f-vectors type D associahedra), A126216 (mirror image).
Cf. A248727 for a relation to f-polynomials of simplices.
Cf. A111785 (contracted partition array, unsigned; see a comment above).
Antidiagonal sums give A005043. - Jordan Tirrell, Jun 01 2017

[[Binomial(n-3, k)*Binomial(n+k-1, k)/(k+1): k in [0..(n-3)]]: n in [3..12]];  // G. C. Greubel, Nov 19 2018

T:=(n,k)->binomial(n-3,k)*binomial(n+k-1,k)/(k+1): seq(seq(T(n,k),k=0..n-3),n=3..12); # Muniru A Asiru, Nov 24 2018

t[n_, k_] = Binomial[n-3, k]*Binomial[n+k-1, k]/(k+1);
Flatten[Table[t[n, k], {n, 3, 12}, {k, 0, n-3}]][[1 ;; 52]] (* Jean-François Alcover, Jun 16 2011 *)

Q=(1+z-(1-(4*w+2+O(w^20))*z+z^2+O(z^20))^(1/2))/(2*(1+w)*z);for(n=3,12,for(m=1,n-2,print1(polcoef(polcoef(Q,n-2,z),m,w),", "))) \\ Hugo Pfoertner, Nov 19 2018

for(n=3,12, for(k=0,n-3, print1(binomial(n-3,k)*binomial(n+k-1,k)/(k+1), ", "))) \\ G. C. Greubel, Nov 19 2018

[[ binomial(n-3,k)*binomial(n+k-1,k)/(k+1) for k in (0..(n-3))] for n in (3..12)] # G. C. Greubel, Nov 19 2018

G.f. G = G(t, z) satisfies (1+t)*G^2 - z*(1-z-2*t*z)*G + t*z^4 = 0.
T(n, k) = binomial(n-3, k)*binomial(n+k-1, k)/(k+1) for n >= 3, 0 <= k <= n-3.
From Tom Copeland, Nov 03 2008: (Start)
Two g.f.s (f1 and f2) for A033282 and their inverses (x1 and x2) can be derived from the Drake and Barry references.
1. a: f1(x,t) = y = {1 - (2t+1) x - sqrt[1 - (2t+1) 2x + x^2]}/[2x (t+1)] = t x + (t + 2 t^2) x^2 + (t + 5 t^2 + 5 t^3) x^3 + ...
b: x1 = y/[t + (2t+1)y + (t+1)y^2] = y {1/[t/(t+1) + y] - 1/(1+y)} = (y/t) - (1+2t)(y/t)^2 + (1+ 3t + 3t^2)(y/t)^3 +...
2. a: f2(x,t) = y = {1 - x - sqrt[(1-x)^2 - 4xt]}/[2(t+1)] = (t/(t+1)) x + t x^2 + (t + 2 t^2) x^3 + (t + 5 t^2 + 5 t^3) x^4 + ...
b: x2 = y(t+1) [1- y(t+1)]/[t + y(t+1)] = (t+1) (y/t) - (t+1)^3 (y/t)^2 + (t+1)^4 (y/t)^3 + ...
c: y/x2(y,t) = [t/(t+1) + y] / [1- y(t+1)] = t/(t+1) + (1+t) y + (1+t)^2 y^2 + (1+t)^3 y^3 + ...
x2(y,t) can be used along with the Lagrange inversion for an o.g.f. (A133437) to generate A033282 and show that A133437 is a refinement of A033282, i.e., a refinement of the f-polynomials of the associahedra, the Stasheff polytopes.
y/x2(y,t) can be used along with the indirect Lagrange inversion (A134264) to generate A033282 and show that A134264 is a refinement of A001263, i.e., a refinement of the h-polynomials of the associahedra.
f1[x,t](t+1) gives a generator for A088617.
f1[xt,1/t](t+1) gives a generator for A060693, with inverse y/[1 + t + (2+t) y + y^2].
f1[x(t-1),1/(t-1)]t gives a generator for A001263, with inverse y/[t + (1+t) y + y^2].
The unsigned coefficients of x1(y t,t) are A074909, reverse rows of A135278. (End)
G.f.: 1/(1-x*y-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-(x+x*y)/(1-x*y/(1-.... (continued fraction). - Paul Barry, Feb 06 2009
Let h(t) = (1-t)^2/(1+(u-1)*(1-t)^2) = 1/(u + 2*t + 3*t^2 + 4*t^3 + ...), then a signed (n-1)-th row polynomial of A033282 is given by u^(2n-1)*(1/n!)*((h(t)*d/dt)^n) t, evaluated at t=0, with initial n=2. The power series expansion of h(t) is related to A181289 (cf. A086810). - Tom Copeland, Sep 06 2011
With a different offset, the row polynomials equal 1/(1 + x)*Integral_{0..x} R(n,t) dt, where R(n,t) = Sum_{k = 0..n} binomial(n,k)*binomial(n+k,k)*t^k are the row polynomials of A063007. - Peter Bala, Jun 23 2016
n-th row polynomial = ( LegendreP(n-1,2*x + 1) - LegendreP(n-3,2*x + 1) )/((4*n - 6)*x*(x + 1)), n >= 3. - Peter Bala, Feb 22 2017
n*T(n+1, k) = (4n-6)*T(n, k-1) + (2n-3)*T(n, k) - (n-3)*T(n-1, k) for n >= 4. - Fang Lixing, May 07 2019

Missing factor of 2 for expansions of f1 and f2 added by Tom Copeland, Apr 12 2009

A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 5, 5, 0, 1, 9, 21, 14, 0, 1, 14, 56, 84, 42, 0, 1, 20, 120, 300, 330, 132, 0, 1, 27, 225, 825, 1485, 1287, 429, 0, 1, 35, 385, 1925, 5005, 7007, 5005, 1430, 0, 1, 44, 616, 4004, 14014, 28028, 32032, 19448, 4862, 0, 1, 54, 936, 7644, 34398, 91728

Offset: 0

Philippe Deléham, Aug 05 2003

Mirror image of triangle A133336. - Philippe Deléham, Dec 10 2008
From Tom Copeland, Oct 09 2011: (Start)
With polynomials
P(0,t) = 0
P(1,t) = 1
P(2,t) = t
P(3,t) = t + 2 t^2
P(4,t) = t + 5 t^2 +  5 t^3
P(5,t) = t + 9 t^2 + 21 t^3 + 14 t^4
The o.g.f. A(x,t) = {1+x-sqrt[(1-x)^2-4xt]}/[2(1+t)] (see Drake et al.).
B(x,t)= x-t x^2/(1-x)= x-t(x^2+x^3+x^4+...) is the comp. inverse in x.
Let h(x,t) = 1/(dB/dx) = (1-x)^2/(1+(1+t)*x*(x-2)) = 1/(1-t(2x+3x^2+4x^3+...)), an o.g.f. in x for row polynomials in t of A181289. Then P(n,t) is given by (1/n!)(h(x,t)*d/dx)^n x, evaluated at x=0, A = exp(x*h(y,t)*d/dy) y, eval. at y=0, and dA/dx = h(A(x,t),t). These results are a special case of A133437 with u(x,t) = B(x,t), i.e., u_1=1 and (u_n)=-t for n > 1. See A001003 for t = 1. (End)
Let U(x,t) = [A(x,t)-x]/t, then U(x,0) = -dB(x,t)/dt and U satisfies dU/dt = UdU/dx, the inviscid Burgers' equation (Wikipedia), also called the Hopf equation (see Buchstaber et al.). Also U(x,t) = U(A(x,t),0) = U(x+tU,0) since U(x,0) = [x-B(x,t)]/t. - Tom Copeland, Mar 12 2012
Diagonals of A132081 are essentially rows of this sequence. - Tom Copeland, May 08 2012
T(r, s) is the number of [0,r]-covering hierarchies with s segments (see Kreweras). - Michel Marcus, Nov 22 2014
From Yu Hin Au, Dec 07 2019: (Start)
T(n,k) is the number of small Schröder n-paths (lattice paths from (0,0) to (2n,0) using steps U=(1,1), F=(2,0), D=(1,-1) with no F step on the x-axis) that has exactly k U steps.
T(n,k) is the number of Schröder trees (plane rooted tree where each internal node has at least two children) with exactly n+1 leaves and k internal nodes. (End)

			Triangle starts:
  1;
  0,  1;
  0,  1,  2;
  0,  1,  5,  5;
  0,  1,  9, 21, 14;
  ...

Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0 <= n <= 150, flattened.)
Yu Hin (Gary) Au, Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers, arXiv:1912.00555 [math.CO], 2019.
Paul Barry, On the Inverses of a Family of Pascal-Like Matrices Defined by Riordan Arrays, Journal of Integer Sequences, 16 (2013), #13.5.6.
Paul Barry, Three Études on a sequence transformation pipeline, arXiv:1803.06408 [math.CO], 2018.
Paul Barry, Generalized Catalan Numbers Associated with a Family of Pascal-like Triangles, J. Int. Seq., Vol. 22 (2019), Article 19.5.8.
V. Buchstaber and E. Bunkova,Elliptic formal group laws, integral Hirzebruch genera and Kirchever genera,, arXiv:1010.0944 [math-ph], 2010 (see p. 19).
V. Buchstaber and T. Panov, Toric Topology. Chapter 1: Geometry and Combinatorics of Polytopes,, arXiv:1102.1079 [math.CO], 2011-2012 (see p. 41).
G. Chatel, V. Pilaud, Cambrian Hopf Algebras, arXiv:1411.3704 [math.CO], 2014-2015.
T. Copeland, Generators, Inversion, and Matrix, Binomial, and Integral Transforms, 2015.
T. Copeland, Compositional inverse pairs, the Burgers-Hopf equation, and the Stasheff associahedra, 2014.
T. Copeland, Lagrange a la Lah, 2011.
B. Drake, Ira M. Gessel, and Guoce Xin, Three Proofs and a Generalization of the Goulden-Litsyn-Shevelev Conjecture on a Sequence Arising in Algebraic Geometry, J. of Integer Sequences, Vol. 10 (2007), Article 07.3.7.
G. Kreweras, Sur les hiérarchies de segments, Cahiers Bureau Universitaire Recherche Opérationnelle, Cahier 20, Inst. Statistiques, Univ. Paris, 1973, p. 21-22.
G. Kreweras, Sur les hiérarchies de segments, Cahiers du Bureau Universitaire de Recherche Opérationnelle, Institut de Statistique, Université de Paris, #20 (1973). (Annotated scanned copy)
J. Zhou, Quantum deformation theory of the Airy curve and the mirror symmetry of a point, arXiv preprint arXiv:1405.5296 [math.AG], 2014.

Diagonals: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279, A000108, A002054, A002055, A002056, A007160, A033280, A033281.
The diagonals (except for A000007) are also the diagonals of A033282.
Row sums: A001003 (Schroeder numbers).
Cf. A033282, A084938.
Cf. A001003, A008297, A021009, A132081, A133437, A181289.

Table[Boole[n == 2] + If[# == -1, 0, Binomial[n - 3, #] Binomial[n + # - 1, #]/(# + 1)] &[k - 1], {n, 2, 12}, {k, 0, n - 2}] // Flatten (* after Jean-François Alcover at A033282, or *)
Table[If[n == 0, 1, Binomial[n, k] Binomial[n + k, k - 1]/n], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Aug 22 2016 *)

t(n, k) = if (n==0, 1, binomial(n, k)*binomial(n+k, k-1)/n);
tabl(nn) = {for (n=0, nn, for (k=0, n, print1(t(n,k), ", ");); print(););} \\ Michel Marcus, Nov 22 2014

Triangle T(n, k) read by rows; given by [0, 1, 0, 1, 0, 1, ...] DELTA [1, 1, 1, 1, 1, 1, 1, 1, 1, ...] where DELTA is Deléham's operator defined in A084938.
For k>0, T(n, k) = binomial(n-1, k-1)*binomial(n+k, k)/(n+1); T(0, 0) = 1 and T(n, 0) = 0 if n > 0. [corrected by Marko Riedel, May 04 2023]
Sum_{k>=0} T(n, k)*2^k = A107841(n). - Philippe Deléham, May 26 2005
Sum_{k>=0} T(n-k, k) = A005043(n). - Philippe Deléham, May 30 2005
T(n, k) = A108263(n+k, k). - Philippe Deléham, May 30 2005
Sum_{k=0..n} T(n,k)*x^k = A000007(n), A001003(n), A107841(n), A131763(n), A131765(n), A131846(n), A131926(n), A131869(n), A131927(n) for x = 0, 1, 2, 3, 4, 5, 6, 7, 8, respectively. - Philippe Deléham, Nov 05 2007
Sum_{k=0..n} T(n,k)*5^k*(-2)^(n-k) = A152601(n). - Philippe Deléham, Dec 10 2008
Sum_{k=0..n} T(n,k)*(-1)^k*3^(n-k) = A154825(n). - Philippe Deléham, Jan 17 2009
Umbrally, P(n,t) = Lah[n-1,-t*a.]/n! = (1/n)*Sum_{k=1..n-1} binomial(n-2,k-1)a_k t^k/k!, where (a.)^k = a_k = (n-1+k)!/(n-1)!, the rising factorial, and Lah(n,t) = n!*Laguerre(n,-1,t) are the Lah polynomials A008297 related to the Laguerre polynomials of order -1. - Tom Copeland, Oct 04 2014
T(n, k) = binomial(n, k)*binomial(n+k, k-1)/n, for k >= 0; T(0, 0) = 1 (see Kreweras, p. 21). - Michel Marcus, Nov 22 2014
P(n,t) = Lah[n-1,-:Dt:]/n! t^(n-1) with (:Dt:)^k = (d/dt)^k t^k = k! Laguerre(k,0,-:tD:) with (:tD:)^j = t^j D^j. The normalized Laguerre polynomials of 0 order are given in A021009. - Tom Copeland, Aug 22 2016

Typo in a(60) corrected by Michael De Vlieger, Nov 21 2019

A077415 a(n) = n(n+2)(n-2)/3.

Original entry on oeis.org

0, 5, 16, 35, 64, 105, 160, 231, 320, 429, 560, 715, 896, 1105, 1344, 1615, 1920, 2261, 2640, 3059, 3520, 4025, 4576, 5175, 5824, 6525, 7280, 8091, 8960, 9889, 10880, 11935, 13056, 14245, 15504, 16835, 18240, 19721, 21280, 22919, 24640, 26445

Offset: 2

Wolfdieter Lang, Nov 29 2002

a(n) is the number of independent components of a 3-tensor t(a,b,c) which satisfies t(a,b,c)=t(b,a,c) and sum(t(a,a,c),a=1..n)=0 for all c and t(a,b,c)+t(b,c,a)+t(c,a,b)=0, with a,b,c range 1..n. (3-tensor in n-dimensional space which is symmetric and traceless in one pair of its indices and satisfies the cyclic identity.)
Number of standard tableaux of shape (n-1,2,1) (n>=3). - Emeric Deutsch, May 13 2004
Zero followed by partial sums of A028387, starting at n=1. - Klaus Brockhaus, Oct 21 2008
For n>=4, a(n-1) is the number of permutations of 1,2...,n with the distribution of up (1) - down (0) elements 0...0101 (the first n-4 zeros), or, the same, a(n-1) is up-down coefficient {n,5} (see comment in A060351). - Vladimir Shevelev, Feb 14 2014
For n>=3, a(n) equals the second immanant of the (n-1) X (n-1) tridiagonal matrix with 2's along the main diagonal, and 1's along the superdiagonal and the subdiagonal. - John M. Campbell, Jan 08 2016

G. C. Greubel, Table of n, a(n) for n = 2..10000
Mark Roger Sepanski, On Divisibility of Convolutions of Central Binomial Coefficients, Electronic Journal of Combinatorics, 21 (1) 2014, #P1.32.
Index entries for linear recurrences with constant coefficients, signature (4,-6,4,-1).

Cf. A000292, A028387 (first differences), A033275 (partial sums), A060351, A077414, A084990.

[n*(n+2)*(n-2)/3: n in [2..50]]; /* or */ I:=[0,5,16,35]; [n le 4 select I[n] else 4*Self(n-1)-6*Self(n-2)+4*Self(n-3)-Self(n-4): n in [1..50]]; // Vincenzo Librandi, Jan 09 2016

seq((n^3-4*n)/3, n=2..35); # Zerinvary Lajos, Jan 20 2007

Print[Table[Sum[(-1)^i*2^(n-2*i-1)*Binomial[n-i-1, i]*(n-2*i-2), {i, 0, Floor[(n-1)/2]}], {n, 2, 100}]] ;  (* John M. Campbell, Jan 08 2016 *)
LinearRecurrence[{4, -6, 4, -1}, {0, 5, 16, 35}, 50] (* Vincenzo Librandi, Jan 09 2016 *)
Table[n*(n + 2)*(n - 2)/3, {n, 2, 50}] (* G. C. Greubel, Jan 18 2018 *)

{a=0; print1(a,","); for(n=1, 42, print1(a=a+n+(n+1)^2, ","))} \\ Klaus Brockhaus, Oct 21 2008

concat(0, Vec(x^3*(5-4*x+x^2)/(1-x)^4 + O(x^100))) \\ Altug Alkan, Jan 08 2015

a(n) = n*(n+2)*(n-2)/3 = A077414(n) - binomial(n+2,3) = A077414(n) - A000292(n-1).
G.f.: x^3*(5 - 4*x + x^2)/(1-x)^4.
a(n) = A084990(n-1) - 1. - Reinhard Zumkeller, Aug 20 2007
a(n) = Sum_{i=0..floor((n-1)/2)} (-1)^i * 2^(n-2*i-1) * binomial(n-i-1, i) * (n-2*i-2). - John M. Campbell, Jan 08 2016
From Amiram Eldar, Jan 06 2021: (Start)
Sum_{n>=3} 1/a(n) = 11/32.
Sum_{n>=3} (-1)^(n+1)/a(n) = 5/32. (End)
E.g.f.: x*(1 + exp(x)*(x^2 + 3*x - 3)/3). - Stefano Spezia, Mar 06 2024

A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 5, 0, 1, 9, 5, 0, 1, 14, 21, 0, 1, 20, 56, 14, 0, 1, 27, 120, 84, 0, 1, 35, 225, 300, 42, 0, 1, 44, 385, 825, 330, 0, 1, 54, 616, 1925, 1485, 132, 0, 1, 65, 936, 4004, 5005, 1287, 0, 1, 77, 1365, 7644, 14014, 7007, 429, 0, 1, 90, 1925, 13650, 34398

Offset: 0

Emeric Deutsch, May 29 2005

Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.

Related to the number of certain non-crossing partitions for the root system A_n. Cf. p. 12, Athanasiadis and Savvidou. Diagonals are A033282/A086810. Also see A132081 and A100754.- Tom Copeland, Oct 19 2014

			T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
Triangle begins:
1;
0;
0,1;
0,1;
0,1,2;
0,1,5;
0,1,9,5

Indranil Ghosh, Rows 0..100, flattened
C. Athanasiadis and C. Savvidou, The local h-vector of the cluster subdivision of a simplex, arXiv preprint arXiv:1204.0362, 2012
F. R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., 204 (1999) 73-112.
Robert Coquereaux and Jean-Bernard Zuber, Counting partitions by genus. II. A compendium of results, arXiv:2305.01100 [math.CO], 2023. See p. 10.

Cf. A005043, A033275, A033276.

Columns: A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A005043 (row sums), A033282, A086810

G:=(1+z-sqrt((1-z)^2-4*t*z^2))/2/z/(1+t*z): Gser:=simplify(series(G,z=0,18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser,z^n) od: for n from 0 to 16 do seq(coeff(t*P[n],t^k),k=1..1+floor(n/2)) od; # yields sequence in triangular form
A108263 := (n,k) -> binomial(n-k-1,n-2*k)*binomial(n,k)/(n-k+1);
seq(print(seq(A108263(n,k),k=0..ceil((n-1)/2))),n=0..8); # Peter Luschny, Sep 25 2014

T[n_,k_]:=Binomial[n-k-1,n-2k]*Binomial[n,k]/(n-k+1); Flatten[Table[T[n,k],{n,0,11},{k,0,Ceiling[(n-1)/2]}]] (* Indranil Ghosh, Feb 20 2017 *)

G.f. G=G(t, z) satisfies z*(1+t*z)*G^2 - (1+z)*G + 1 = 0.

T(n, k) = A086810(n-k, k). - Philippe Deléham, May 30 2005

A033276 Number of diagonal dissections of an n-gon into 4 regions.

Original entry on oeis.org

0, 14, 84, 300, 825, 1925, 4004, 7644, 13650, 23100, 37400, 58344, 88179, 129675, 186200, 261800, 361284, 490314, 655500, 864500, 1126125, 1450449, 1848924, 2334500, 2921750, 3627000, 4468464, 5466384, 6643175, 8023575, 9634800, 11506704, 13671944

Offset: 5

N. J. A. Sloane

Number of standard tableaux of shape (n-4,2,2,2) (n>=6). - Emeric Deutsch, May 20 2004

Number of short bushes with n+2 edges and 4 branch nodes (i.e. nodes with outdegree at least 2). A short bush is an ordered tree with no nodes of outdegree 1. Example: a(6)=14 because the only short bushes with 8 edges and 4 branch nodes are the fourteen full binary trees with 8 edges. Column 4 of A108263. - Emeric Deutsch, May 29 2005

Vincenzo Librandi, Table of n, a(n) for n = 5..1000
David Beckwith, Legendre polynomials and polygon dissections?, Amer. Math. Monthly, Vol. 105, No. 3 (1998), 256-257.
Frank R. Bernhart, Catalan, Motzkin and Riordan numbers, Discr. Math., Vol. 204, No. 1-3 (1999), 73-112.
Ronald C. Read, On general dissections of a polygon, Aequat. math. 18 (1978), 370-388, Table 1.
Index entries for linear recurrences with constant coefficients, signature (7,-21,35,-35,21,-7,1).

Cf. A033275, A108263.

[(Binomial(n+2,3)*Binomial(n-3,3))/4: n in [5..50]]; // Vincenzo Librandi, Mar 15 2014

Table[(Binomial[n+2,3]Binomial[n-3,3])/4,{n,5,40}] (* or *) LinearRecurrence[ {7,-21,35,-35,21,-7,1},{0,14,84,300,825,1925,4004},40] (* Harvey P. Dale, Mar 13 2014 *)
CoefficientList[Series[x (14 - 14 x + 6 x^2 - x^3)/(1 - x)^7, {x, 0, 40}], x] (* Vincenzo Librandi, Mar 15 2014 *)

a(n) = binomial(n+2, 3)*binomial(n-3, 3)/4.
G.f.: x^6*(14-14x+6x^2-x^3)/(1-x)^7. - Emeric Deutsch, May 29 2005
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=6} 1/a(n) = 109/1225.
Sum_{n>=6} (-1)^n/a(n) = 192*log(2)/35 - 4582/1225. (End)

More terms from Vincenzo Librandi, Mar 15 2014

A299120 a(n) = (n-1)(n-2)(n+3)*(n+2)/12.

Original entry on oeis.org

1, 0, 0, 5, 21, 56, 120, 225, 385, 616, 936, 1365, 1925, 2640, 3536, 4641, 5985, 7600, 9520, 11781, 14421, 17480, 21000, 25025, 29601, 34776, 40600, 47125, 54405, 62496, 71456, 81345, 92225, 104160, 117216, 131461, 146965, 163800, 182040, 201761, 223041

Offset: 0

Juri-Stepan Gerasimov, Feb 03 2018

Colin Barker, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A001263, A033275, A077415, A299146, A299198.

List([0..10^3], n->n^4/12+n^3/6-7*n^2/12-2*n/3+1); # Muniru A Asiru, Feb 04 2018

[n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1: n in [0..40]];

seq(n^4/12+n^3/6-7*n^2/12-2*n/3+1, n=0..10^3); # Muniru A Asiru, Feb 04 2018

Rest@ CoefficientList[Series[(1 - 5 x + 10 x^2 - 5 x^3 + x^4)/(1 - x)^5, {x, 0, 41}], x] (* Michael De Vlieger, Feb 10 2018 *)
f[n_] := n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1; Array[f, 40, 0] (* or *)
LinearRecurrence[{5, -10, 10, -5, 1}, {1, 0, 0, 5, 21}, 40] (* Robert G. Wilson v, Mar 12 2018 *)

Vec((1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5 + O(x^50)) \\ Colin Barker, Feb 05 2018

a(n) = n^4/12 + n^3/6 - 7*n^2/12 - 2*n/3 + 1 = (n-1)*(n-2)*(n+3)*(n+2)/12.
From Colin Barker, Feb 05 2018: (Start)
G.f.: (1 - 5*x + 10*x^2 - 5*x^3 + x^4) / (1 - x)^5.
a(n) = 5*a(n-1) - 10*a(n-2) + 10*a(n-3) - 5*a(n-4) + a(n-5) for n>5. (End)
a(n) = A033275(n+2) for n > 1. - Georg Fischer, Oct 09 2018
From Amiram Eldar, Jan 12 2021: (Start)
Sum_{n>=3} 1/a(n) = 43/150.
Sum_{n>=3} (-1)^(n+1)/a(n) = 16*log(2)/5 - 154/75. (End)
E.g.f.: exp(x)*(12 - 12*x + 6*x^2 + 8*x^3 + x^4)/12. - Stefano Spezia, Feb 21 2024

Edited by Wolfdieter Lang, Apr 06 2018

A104473 a(n) = binomial(n+2,2)*binomial(n+6,2).

Original entry on oeis.org

15, 63, 168, 360, 675, 1155, 1848, 2808, 4095, 5775, 7920, 10608, 13923, 17955, 22800, 28560, 35343, 43263, 52440, 63000, 75075, 88803, 104328, 121800, 141375, 163215, 187488, 214368, 244035, 276675, 312480, 351648, 394383, 440895, 491400, 546120, 605283, 669123

Offset: 0

Zerinvary Lajos, Apr 18 2005

			a(0) = C(0+2,2)*C(0+6,2) = C(2,2)*C(6,2) = 1*15 = 155.
a(6) = 1*3*5 + 2*4*6 + 3*5*7 + 4*6*8 + 5*7*9 + 6*8*10 + 7*9*11 = 1848.

Vincenzo Librandi, Table of n, a(n) for n = 0..1000
Index entries for linear recurrences with constant coefficients, signature (5,-10,10,-5,1).

Cf. A000217, A000579, A033275, A034856, A062264.

Subsequence of A085780.

[Binomial(n+2, 2)*Binomial(n+6, 2): n in [0..50]]; // Vincenzo Librandi, Apr 28 2014

f[n_] := Binomial[n + 2, 2] Binomial[n + 6, 2]; Table[f[n], {n,0,40}] (* Robert G. Wilson v, Apr 20 2005 *)
CoefficientList[Series[3 (5-4*x+x^2)/(1-x)^5, {x,0,40}], x] (* Vincenzo Librandi, Apr 28 2014 *)

a(n)=binomial(n+2,2)*binomial(n+6,2) \\ Charles R Greathouse IV, Jun 07 2013

def A104473(n): return binomial(n+2,2)*binomial(n+6,2)
print([A104473(n) for n in range(51)]) # G. C. Greubel, Mar 05 2025

a(n) = (1/4)*(n+1)*(n+2)*(n+5)*(n+6).
a(n) = A034856(n+2)^2 - 1. - J. M. Bergot, Dec 14 2010
G.f.: 3*(5-4*x+x^2)/(1-x)^5. - Colin Barker, Sep 21 2012
a(n) = Sum_{i=1..n+1} i*(i+2)*(i+4). - Bruno Berselli, Apr 28 2014
a(n) = A000217(n)*A000217(n+4) = 3*A033275(n+4). - R. J. Mathar, Nov 29 2015
From Amiram Eldar, Aug 30 2022: (Start)
Sum_{n>=0} 1/a(n) = 43/450.
Sum_{n>=0} (-1)^n/a(n) = 16*log(2)/15 - 154/225. (End)
From G. C. Greubel, Mar 05 2025: (Start)
a(n) = 90*A000579(n+6)/A000279(n+3).
E.g.f.: (1/4)*(60 + 192*x + 114*x^2 + 20*x^3 + x^4)*exp(x). (End)

A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 0, 2, 5, 3, 0, 3, 9, 12, 6, 0, 4, 13, 21, 22, 10, 0, 5, 17, 30, 38, 35, 15, 0, 6, 21, 39, 54, 60, 51, 21, 0, 7, 25, 48, 70, 85, 87, 70, 28, 0, 8, 29, 57, 86, 110, 123, 119, 92, 36, 0, 9, 33, 66, 102, 135, 159, 168, 156, 117, 45

Offset: 0

Paul Curtz, Mar 05 2023

The main diagonal is A002414.
The first upper diagonal is A160378(n+1).
The antidiagonals sums are A034827(n+2).
b(n) = (A034827(n+3) = 0, 2, 10, 30, 70, ...) - (A002414(n) = 0, 1, 9, 30, 70, ...) = 0, 1, 1, 0, 0, 5, 21, 56, ... .
b(n+2) = A299120(n). b(n+4) = A033275(n). b(n+4) - b(n) = A002492(n).

			The rows are
  0, 0,  1,  3,  6,  10,  15,  21, ...   = A161680
  0, 1,  5, 12, 22,  35,  51,  70, ...   = A000326
  0, 2,  9, 21, 38,  60,  87, 119, ...   = A005476
  0, 3, 13, 30, 54,  85, 123, 168, ...   = A022264
  0, 4, 17, 39, 70, 110, 159, 217, ...   = A022266
  ... .
Columns: A000004, A001477, A016813, A017197=3*A016777, 2*A017101, 5*A016873, 3*A017581, 7*A017017, ... (coefficients from A026741).
Difference between two consecutive rows: A000290. Hence A143844.
This square array read by antidiagonals leads to the triangle
  0
  0   0
  0   1   1
  0   2   5   3
  0   3   9  12   6
  0   4  13  21  22  10
  0   5  17  30  38  35  15
  ... .

Cf. A000004, A000290, A000326, A001477, A002414, A005476, A016777, A016813, A016873, A017017, A017101, A017197, A017581, A022264, A022266, A026741, A034827, A160378, A161680, A360962.

Cf. A002492, A033275, A143844, A299120.

T[n_, k_] := k*((2*n + 1)*k - 1)/2; Table[T[n - k, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Amiram Eldar, Mar 05 2023 *)

a(n) = { my(row = (sqrtint(8*n+1)-1)\2, column = n - binomial(row + 1, 2)); binomial(column, 2) + column^2 * (row - column) } \\ David A. Corneth, Mar 05 2023

# Seen as a triangle:
from functools import cache
@cache
def Trow(n: int) -> list[int]:
    if n == 0: return [0]
    r = Trow(n - 1)
    return [r[k] + k * k if k < n else r[n - 1] + n - 1 for k in range(n + 1)]
for n in range(7): print(Trow(n)) # Peter Luschny, Mar 05 2023

Take successively sequences n*(n-1)/2, n*(3*n-1)/2, n*(5*n-1)/2, ... listed in the EXAMPLE section.
G.f.: y*(x + y)/((1 - y)^3*(1 - x)^2). - Stefano Spezia, Mar 06 2023
E.g.f.: exp(x+y)*y*(2*x + y + 2*x*y)/2. - Stefano Spezia, Feb 21 2024

cp's OEIS Frontend

A033282 Triangle read by rows: T(n, k) is the number of diagonal dissections of a convex n-gon into k+1 regions.

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A086810 Triangle obtained by adding a leading diagonal 1,0,0,0,... to A033282.

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A077415 a(n) = n*(n+2)*(n-2)/3.

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A108263 Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e., nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1.

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A033276 Number of diagonal dissections of an n-gon into 4 regions.

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A299120 a(n) = (n-1)*(n-2)*(n+3)*(n+2)/12.

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A077415 a(n) = n(n+2)(n-2)/3.

A299120 a(n) = (n-1)(n-2)(n+3)*(n+2)/12.

A361226 Square array T(n,k) = k((1+2n)*k - 1)/2; n>=0, k>=0, read by antidiagonals upwards.